Tensor Calculus Research Articles

General Solutions to the Navier-Stokes Equations for Incompressible Flow

<i>Waves are mainly generated by wind via the transfer of wind energy to the water through friction. When the wind subsides, the waves transition into swells and eventually dissipate. Friction plays a crucial role in the generation and dissipation of waves. Numerous wave theories have been developed based on the assumption of inviscid flow, but these theories are inadequate in explaining the transformation of waves into swells. This study addressed these limitations by analytically deriving general solutions to the Navier–Stokes equations. By expressing the velocity field as the product of a solution to the Helmholtz equation and a time-dependent univariate function, the Navier–Stokes equations are decomposed into an ordinary differential equation and the Euler equations, which are solved using tensor calculus. This paper provides solutions for viscous flow with shear currents when applied to the water wave problem. These solutions were validated through their application to the vorticity equation. The decay modulus of water waves was compared with experimental data, showing a significant degree of concordance. In contrast to other wave theories, this study clarified the process through which waves evolve into swells.</i>

Some remarks on the history of Ricci’s absolute differential calculus

The article offers a general account of the genesis of the absolute differential calculus (ADC), paying special attention to its links with the history of differential geometry. In relatively recent times, several historians have described the development of the ADC as a direct outgrowth either of the theory of algebraic and differential invariants or as a product of analytical investigations, thus minimizing the role of Riemann’s geometry in the process leading to its discovery. Our principal aim consists in challenging this historiographical tenet and analyzing the intimate connection between the development of Riemannian geometry and the birth of tensor calculus.

Components of curvature-squared invariants of minimal supergravity in five dimensions

We present for the first time the component structure of the supersymmetric completions for all curvature-squared invariants of five-dimensional, off-shell (gauged) minimal supergravity, including all fermions. This is achieved by using an interplay between superspace and superconformal tensor calculus techniques, and by employing results from arXiv:1410.8682 and arXiv:2302.14295. Our analysis is based on using a standard Weyl multiplet of conformal supergravity coupled to a vector and a linear multiplet compensator to engineer off-shell Poincaré supergravity. We compute all the descendants of the composite linear multiplets that describe gauged supergravity together with the three independent four-derivative invariants. These are the building blocks of the locally superconformal invariant actions. A derivation of the primary equations of motion for minimal gauged off-shell supergravity deformed by an arbitrary combination of these three locally superconformal invariants, is then provided. Finally, all the covariant descendants in the multiplets of equations of motion are obtained by applying a series of Q-supersymmetry transformations, equivalent to successively applying superspace spinor derivatives to the primary equations of motion.

Scalar curvature for metric spaces: Defining curvature for quantum gravity without coordinates

Geometrical properties of spacetime are difficult to study in nonperturbative approaches to quantum gravity like causal dynamical triangulations (CDT), where one uses simplicial manifolds to define the gravitational path integral, instead of Riemannian manifolds. In particular, in CDT one only relies on two mathematical tools, a distance measure and a volume measure. In this paper, we define a notion of scalar curvature, for metric spaces endowed with a volume measure or a random walk, without assuming nor using notions of tensor calculus. Furthermore, we directly define the Ricci scalar, without the need of defining and computing the Riemann or the Ricci tensor . For this, we make use of quantities, like the surface of a geodesic sphere, or the return probability of scalar diffusion processes, that can be computed in these metric spaces, as in a Riemannian manifold, where they receive scalar curvature contributions. Our definitions recover the classical results of scalar curvature when the sets are Riemannian manifolds. We propose two methods to compute the scalar curvature in these spaces, and we compare their features in natural implementations in discrete spaces. The defined generalized scalar curvatures are easily implemented on discrete spaces, like graphs. We present the results of our definitions on random triangulations of a 2D sphere and plane. Additionally, we show the results of our generalized scalar curvatures on the quantum geometries of 2D CDT, where we find that all our definitions indicate a flat ground state of the gravitational path integral. Published by the American Physical Society 2024

Linearity Study of Color Measurement Methods in Digital Image Colorimetry

Color metrology is steadily developing in the interests of fundamental and applied science, improving industrial production processes. The author considers the problem of ensuring metrological traceability of color measurements under natural conditions.Digital image colorimetry methods demonstrate good linearity and precision in relatively small application ranges in studies of same type and mainly non-luminous objects. For complex scenes with a wide range of brightnesses, nonlinear functional color rendering models are used that are focused on visual perception, peak brightness of the monitor and reference viewing conditions, which complicates the processing of measurement results and reduces a confidence interval.The described method of measuring the chromaticity coordinates of an object by its digital image is based on tensor calculus and consists of multiple registration of an object and sets of reference samples with different exposure times, building calibration dependencies forming a multidimensional colorimetric scale in color space and determining chromaticity coordinates as points of intersection of color tensors with the plane of the color locus. In order to ensure accuracy and reliability of the measurement results in the software and hardware environments, a validation study of the performance indicators of this method was carried out, one of which is linearity. An experiment based on comparing the developed method with the reference measurement method showed the effectiveness of the developed wide-range measurement method with good linearity (correlation coefficient R=0.97… 0.99, residual relative standard deviation s0≈1–2%) when adjusting from the subjective factor and display device.The experimental materials may be useful to metrologists solving problems of determining the metrological traceability of the color of digital images in media. The article will be of interest to a wide range of readers with professional interests related to the colorimetry of digital images.

On the first eigenvalue of the Laplacian for polygons

A 2006 conjecture of Antunes and Freitas is addressed connecting the scaling-invariant polygonal isoperimetric and principal frequency deficits for triangles. This yields a quantitative polygonal Faber–Krahn inequality for triangles with an explicit constant. Furthermore, a problem mentioned in the 1951 book “Isoperimetric Inequalities In Mathematical Physics” by Pólya and Szegö is addressed: a formula is given for the principal frequency of a triangle. Moreover, a space of polygons is constructed for the classical Pólya and Szegö problem: in 1947, Pólya proved that if n = 3, 4 the regular polygon Pn minimizes the principal frequency of an n-gon with given area α > 0 and suggested that the same holds when n ≥ 5. In 1951, Pólya and Szegö discussed the possibility of counterexamples. This paper constructs explicit (2n − 4)–dimensional polygonal manifolds M(n,α) and proves the existence of a computable N ≥ 5 such that for all n ≥ N, the admissible n-gons are given via M(n,α)and there exists an explicit set An(α)⊂M(n,α) such that Pn has the smallest principal frequency among n-gons in An(α). Inter-alia when n ≥ 3, a formula is proved for the principal frequency of a convex P∈M(n,α)in terms of an equilateral n-gon with the same area; and, the set of equilateral polygons is proved to be an (n − 3)–dimensional submanifold of the (2n − 4)–dimensional manifold M(n,α)near Pn. The techniques involve a partial symmetrization, tensor calculus, the spectral theory of circulant matrices, and W2,p/BMO estimates. Last, an application is given in the context of electron bubbles.

Higher-order chain rules for tensor fields, generalized Bell polynomials, and estimates in Orlicz-Sobolev-Slobodeckij and total variation spaces

We describe higher-order chain rules for multivariate functions and tensor fields. We estimate Sobolev-Slobodeckij norms, Musielak-Orlicz norms, and the total variation seminorms of the higher derivatives of tensor fields after a change of variables and determine sufficient regularity conditions for the coordinate change. We also introduce a novel higher-order chain rule for composition chains of multivariate functions that is described via nested set partitions and generalized Bell polynomials; it is a natural extension of the Faà di Bruno formula. Our discussion uses the coordinate-free language of tensor calculus and includes Fréchet-differentiable mappings between Banach spaces.

Local vector measures

Consider a BV function on a Riemannian manifold. What is its differential? And what about the Hessian of a convex function? These questions have clear answers in terms of (co)vector/matrix valued measures if the manifold is the Euclidean space. In more general curved contexts, the same objects can be perfectly understood via charts. However, charts are often unavailable in the less regular setting of metric geometry, where still the questions make sense.In this paper we propose a way to deal with this sort of problems and, more generally, to give a meaning to a concept of ‘measure acting in duality with sections of a given bundle’, loosely speaking. Despite the generality, several classical results in measure theory like Riesz's and Alexandrov's theorems have a natural counterpart in this setting. Moreover, as we are going to discuss, the notions introduced here provide a unified framework for several key concepts in nonsmooth analysis that have been introduced more than two decades ago, such as: Ambrosio-Kirchheim's metric currents, Cheeger's Sobolev functions and Miranda's BV functions.Not surprisingly, the understanding of the structure of these objects improves with the regularity of the underlying space. We are particularly interested in the case of RCD spaces where, as we will argue, the regularity of several key measures of the type we study nicely matches the known regularity theory for vector fields, resulting in a very effective theory.We expect that the notions developed here will help creating stronger links between differential calculus in Alexandrov spaces (based on Perelman's DC charts) and in RCD ones (based on intrinsic tensor calculus).

Intrinsic Nonlinear Elasticity: An Exterior Calculus Formulation

In this paper, we formulate the theory of nonlinear elasticity in a geometrically intrinsic manner using exterior calculus and bundle-valued differential forms. We represent kinematics variables, such as velocity and rate of strain, as intensive vector-valued forms, while kinetics variables, such as stress and momentum, as extensive covector-valued pseudo-forms. We treat the spatial, material and convective representations of the motion and show how to geometrically convert from one representation to the other. Furthermore, we show the equivalence of our exterior calculus formulation to standard formulations in the literature based on tensor calculus. In addition, we highlight two types of structures underlying the theory: first, the principal bundle structure relating the space of embeddings to the space of Riemannian metrics on the body and how the latter represents an intrinsic space of deformations and second, the de Rham complex structure relating the spaces of bundle-valued forms to each other.

Addendum to “fourth-order tensor calculus operations and application to continuum mechanics”

The authors provided a novel explicit expression for the dot derivative of an isotropic tensor function in the work by Kellermann et al. An alternate proof is provided in this addendum without the power series assumption.

Fourth-order tensor calculus operations and application to continuum mechanics

This paper provides a presentation of differential calculus involving fourth-order tensors in consideration of the cyclic class-based tensor operations introduced in an earlier paper on fourth-order tensor algebraic operations. Three classes of cyclic second-order tensor operators, denoted by dot, cross, and star, with separate definitions for double contraction, quadruple contraction, and tensor products were recognized. The present paper examines the differentiation of second-order isotropic tensor functions with respect to a second-order tensor, taking into account the three classes of second-order tensor operators. Product rules, chain rules, and identities necessary to maintain consistency are defined. In combining the proposed calculus rules with the second-order tensor operators, a comprehensive set of proofs for the power derivatives is first presented. A closed-form solution for the second-order tensor derivative of an isotropic tensor function is also detailed. Some of the new solutions are an outcome of capabilities offered by new operators and identities provided in the literature. The paper is arranged so as to be a useful reference work with regard to the mathematics of fourth-order tensors in absolute notation, particularly for application in the field of continuum mechanics in engineering.

Invariance of Maxwell’s equations to Lorentz transformations: conceptual bridges through derivation as an alternative to tensorial calculus

Na tradição do campo disciplinar da Física, e nos respectivos manuais didáticos, a invariância das equações de Maxwell por transformações de Lorentz é usualmente apresentada via cálculo tensorial, do que decorre que seu ensino seja postergado até que (supostamente) os estudantes-alvo dominem esse formalismo. Isso pode acarretar, em disputas curriculares as mais disformes, que muitos sequer tenham acesso instrucional a esse tópico, com relevantes implicações para o conhecimento avançado da Física Moderna e Contemporânea. Desse diagnóstico, desenvolvemos uma alternativa de se proceder àquela apresentação usando-se apenas operações básicas da derivação, como a regra da cadeia. Na direção dos processos de ensino e aprendizagem associáveis, sustentamo-nos no aporte teórico das pontes conceituais, por nós desenvolvido a partir da epistemologia cognitiva de objeto homólogo de Gérard Vergnaud. Esse artifício de congruência referencial buscou estabelecer a transitabilidade das maneiras, sem hierarquização, mas com evidente implicação de que a alternativa demonstrada poderia facilmente ser introduzida em disciplinas iniciais de Eletromagnetismo, tipicamente realizadas no primeiro ciclo dos cursos de Física e Engenharia, contribuindo para a retomada progressiva do assunto em módulos avançados.

A Vector Tensor Calculus Description of a Euclidean Space

We present a tensor description of Euclidean spaces that emphasizes the use of geometric vectors which leads to greater geometric insight and a higher degree of organization in analytical expressions. We demonstrate the effectiveness of the approach by proving a number of integral identities with vector integrands. The presented approach may be aptly described as absolute vector calculus or as vector tensor calculus.

Tensor Calculus in Digital Colorimetry

Any object can have many implementations in the form of digital images and any digital image can be processed many times increasing or decreasing accuracy and reliability. Digital colorimetry faces the need to work out issues of ensuring accuracy, metrological traceability and reliability. The purpose of this work was to generalize approaches to the description of multidimensional quantized spaces and show the possibilities of their adaptation to digital colorimetry. This approach will minimize the private and global risks in measurements.For color identification digital colorimetry uses standard color models and spaces. Most of them are empirical and are improved during the transition from standard to real observation conditions taking into account the phenomena of vision and the age of observers. From the point of view of measurement, a digital image can be represented by a combinatorial model of an information and measurement channel with the appearance of the phenomenon of a color covariance hypercube requiring a significant amount of memory for data storage and processing. The transition from the covariance hypercube to high-dimensional matrices and tensors of the first, second and higher ranks provides the prospect of optimizing the color parameters of a digital image by the criterion of information entropy.Tensor calculus provides opportunities for expanding the dynamic range in color measurements describing multidimensional vector fields and quantized spaces with indexing tensors and decomposing them into matrices of low orders.The proposed complex approach based on tensor calculus. According to this approach the color space is a set of directed vector fields undergoing sampling, quantization and coding operations. Also it is a dynamic open system exchanging information with the environment at a given level and to identify color with specified levels of accuracy, reliability, uncertainty and entropy.

The Theory of Magnitudes

This is a way to look at a photos of stars and make sense to get a value that shows the closest star: in the vicinity: It uses star’s apparent magnitudes to get theses inverse magnitudes and to get a nearby star of the largest ratio ; of the inverse apparent magnitude divided by the distance. Now the beginning reference star obeys this ratio . For the greater the distance the lower the value. And we want the highest value . The greater the brightness enlarges and the greater the distance makes the value lessons. After trying to use aspects of General Relativity’s Tensor Calculus, l found l could not find my formulas to fit the data . In an original way l used the inverse Apparent Magnitudes in Astronomical to get a consistant formula that is data proven with general ideas from Global analysis. It works with Star charts/photos. Before l even tried using the inverse of the determinates of linear algebra. But my attempts were to no avail. Therefore l became inspired by basic formulas in General Relativity but that did not work. Using Global Analysis in an new l succeeded. Keywords Using my new formula l can obtain the distance between a reference star, using its inverse apparent magnitude, l obtain the distance to from another star using certain restrictions which the measurement in centimeters to this other star from its reference star nearby.

Vibration Analysis of Curved Panel Subjected to Internal Pressure and Axial Compression

This paper discusses the change of the natural frequencies and the corresponding mode shapes of an isotropic metallic curved panel subjected to combined internal pressure and axial compression. Tensor calculus is employed to devise the differential operator matrices of the geometrical relation in an orthogonal curvilinear reference system. In the framework of the Carrera unified formulation and by applying the finite element method, the principle of virtual work is used to describe the free vibration of a cylindrical shell subjected to progressive external loading conditions as a virtual procedure of the experimental test vibration correlation technique. Finally, numerical examples are provided, and the effects of the material and geometric characteristics are evaluated. The results show that crossing and veering phenomena between natural frequencies appear when varying the curvature of the panel and the value of the inner pressure, and those effects have to be taken into account for a proper design of this type of structure.

Tensor of Order Two and Geometric Properties of 2D Metric Space

A 2D metric space has a limited number of properties through which it can be described. This metric space may comprise objects such as a scalar, a vector, and a rank-2 tensor. The paper provides a comprehensive description of relations between objects in 2D space using the matrix product of vectors, geometric product, and dot product of complex numbers. These relations are also an integral part of the Lagrange’s identity. The entire structure of derived theoretical relationships describing properties of 2D space draws on the Lagrange’s identity. The description of how geometric algebra and tensor calculus are interconnected is given here in a comprehensive and essentially clear manner, which is the main contribution of this paper. A new term in this regard is the total geometric and matrix product, which—in a simple manner—predetermines and defines the existence of differential relations such as the gradient, the divergence, and the curl of a vector field. In addition, geometric interpretation of tensors is pointed out, expressed through angular parameters known from the literature as a tensor glyph. This angular interpretation of the tensor has an unequivocal analytical form, and the paper shows how it is linked to the classical tensor denoted by indices.

Riemannian Calculus of Variations Using Strongly Typed Tensor Calculus

In this paper, the notion of strongly typed language will be borrowed from the field of computer programming to introduce a calculational framework for linear algebra and tensor calculus for the purpose of detecting errors resulting from inherent misuse of objects and for finding natural formulations of various objects. A tensor bundle formalism, crucially relying on the notion of pullback bundle, will be used to create a rich type system with which to distinguish objects. The type system and relevant notation is designed to “telescope” to accommodate a level of detail appropriate to a set of calculations. Various techniques using this formalism will be developed and demonstrated with the goal of providing a relatively complete and uniform method of coordinate-free computation. The calculus of variations pertaining to maps between Riemannian manifolds will be formulated using the strongly typed tensor formalism and associated techniques. Energy functionals defined in terms of first-order Lagrangians are the focus of the second half of this paper, in which the first variation, the Euler–Lagrange equations, and the second variation of such functionals will be derived.

Tensor Quantization: High-Dimensional Data Compression

Quantization is an important technique to transform the input sample values from a large set (or a continuous range) into the output sample values in a small set (or a finite set). It has been applied broadly for lossy-data compression, pattern recognition, probability density estimation, and clustering. Vector quantization (VQ) is a prevalent image-compression technique, which treats image matrices as stretched vectors and then finds the representative stretched vectors accordingly for a given image data set. One can use tensor data representation to directly characterize the original two-dimensional image data rather than stretch the image matrix into a long vector so as to destroy the original two-dimensional data structure. In this work, we propose a new tensor quantization (TQ) framework which does not need to reduce the dimensionality of the original image data and destroy the original two-dimensional spatial relationship among data; these two serious drawbacks of vector quantization are well known. We first present tensor calculus and then propose a new parallel tensor-inversion algorithm for TQ thereupon. We also establish the pertinent theoretical proof to justify that our proposed new TQ approach is superior to the existing VQ approach especially as the image dimension becomes large. Finally, numerical experiments to evaluate the image-compression performances of VQ and TQ are demonstrated and their corresponding computational-complexities are also compared.

Iterator-Based Design of Generic C++ Algorithms for Basic Tensor Operations

Numerical tensor calculus has recently gained increasing attention in many scientific fields including quantum computing and machine learning which contain basic tensor operations such as the pointwise tensor addition and multiplication of tensors. We present a C++ design of multi-dimensional iterators and iterator-based C++ functions for basic tensor operations using mode-specific iterators only, simplifying the implementation of algorithms with recursion and multiple loops. The proposed C++ functions are designed for dense tensor and subtensor types with any linear storage format, mode and dimensions. We demonstrate our findings with Boost's latest uBlas tensor extension and discuss how other C++ frameworks can utilize our proposal without modifying their code base. Our runtime measurements show that C++ functions with iterators can compute tensor operations at least as fast as their pointer-based counterpart.